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Complex filter factorizations - continued

Continuing from this rather silly trivial question factoring real valued filters into shorter complex ones, hoping this won't be as trivial. If we modify it a bit: $$z_0 = e^{2\pi i / 8}$$ and ...
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23 views

Complex filter factorizations

There is a famous low pass filter $[1,2,1]$ in signal processing which can be factored in the sense of a convolution product over the real numbers : $[1,1] * [1,1]$. This is the only way to do it over ...
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1answer
60 views

Signal processing : future values prediction

Let $f : \mathbb{R}^+ \rightarrow \mathbb{R} $ be a continuous function. Do you have some references (books or online resource) about techniques that allow to predict $f(x_{n+1})$, knowing $f(x_0), ...
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1answer
20 views

Second Order Damped System (Harmonic Oscillator)

I'm currently working on an electrical engineering problem involving an ideal operational amplifier. The system itself is governed by the transfer function, which relates output to input: ...
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37 views

Extracting a Cosine Function from a Linear Combination of Cosines

I have a frequency modulated signal which must contain only $ g(t)=B.\cos(\omega(t).t+\phi)$, but it gets the form as below $$ ...
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1answer
38 views

Inverse $Z$ transform of $\frac{1}{z-a}$

I don't really get what's happening here and I haven't been able to find a single example on how to get the inverse $Z$-transform of $\frac{1}{z-a}$. Can anyone show the way?
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1answer
46 views

is a “non-causal” system “memory”?

Is their a relationship between non-causality and memory? for example: is the system $Y(t) = X(t+1)$ memory or memory-less. I got confused because the memory system is defined to depend only on the ...
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1answer
81 views

Calculating convolution integral analytically

How can i compute convolution integral analytically, without using graphs. I hate using graphs, shiftings which are error prone. If this is possible can you explain what way i must follow? For ...
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1answer
42 views

How to mathematically model noise?

In my project I have to perform analysis of noise effect in certain signal. I am just wondering how is noise formally described? Up to now I always simulate a noisy signal using MATLAB in an additive ...
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35 views

Relation between Covariance and Energy of a random signal

Let's say I have the below random signal: $Y[n]=[y(n)y(n−1)y(n−2)....y(1)]$ I have two random variables now: The first one $X_1$ which express the maximum eigenvalue of the covariance matrix of $Y$. ...
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15 views

Does inverse of all Fouriers transforms have a corresponding function in time domain?

I am trying to cancel out the following transfer function of a system: $$\frac{( 1 - e^{(i*k*T)} ) }{ (i*k)}$$ I thought it would work if I find the inverse Fourier transform of $$\frac{ (i*k)}{( 1 ...
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2answers
64 views

Why is $\int e^{-t}u(t) dt = (1-e^{-t})u(t) + Constant$?

How do you solve $\int e^{-t}u(t) dt $? In which u(t) is the unit step function. $\int e^{-t}u(t) dt = (1-e^{-t})u(t) + Constant$ But what are the intermediate steps? Unit step u(t) = ...
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19 views

explanation of correlation of stationary stochastic processes

I have some questions about correlation in stationary stochastic processes. I know that the expectation of a random variable is $E(x)=\int_{-\infty}^{+\infty} a ...
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3answers
55 views

Why is the convolution output in terms of 't' not $\tau$?

The convolution integral is defined as: $$y(t) = (h * x)(t) = \int^{+\infty}_{-\infty} h(\tau). x(t-\tau)\ d\tau$$ where $h(t)$ and $x(t)$ are functions in terms of time. Why is $y$ in terms of ...
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2answers
39 views

Do decaying exponential signals have finite energy?

"A signal that decays exponentially has finite energy, so, it is also an energy signal." http://www.songho.ca/dsp/signal/signals.html I don't quite get how that can be true. Energy of a ...
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18 views

How to generate random noise with given parameters?

I'm studying optical fibers and trying to analyze how the core radius fluctuations along the fiber length affect the performance of optical fibers. How can I generate random noise dr which satisfies ...
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21 views

Is there a name for this “simplified” Volterra series?

Consider a nonlinear, time-invariant system of the following form: $g(t) = \left[h_1(t) \ast f(t)^1\right] + \left[h_2(t) \ast f(t)^2\right] + \left[h_3(t) \ast f(t)^3\right] + ...$ where $\ast$ ...
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1answer
39 views

Transforming a sawtooth into a sinus with one parameter

Can you help me in finding the analytical expression of a function $f_\alpha(\theta)$, with one parameter $\alpha=(0,1)$ by which one can continously transform a sawtooth curve into a sinus? With ...
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9 views

Estimate function given PDF and covariance

Let's say $h(x)$, random variable, represents the height of a surface, with x being the usual x-axis. The probability distribution function is: $P(h) = Ke^{-\frac{h^2}{2s^2}}$ is Gaussian, where $K$ ...
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20 views

Math required to solve Fourier Transform for periodic functions?

In my Signals & Systems course my professor said that although it's possible to solve the standard Fourier transform for periodic signals, it requires advanced math beyond what we've learned, so ...
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1answer
21 views

Assist on finding the impulse response fom a simple LTI graph

Could you please advise on the impulse response from this $LTI$ graph? I need to plot it and find if it is stable and causal... LTI system with response to unitary step ($t$):
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1answer
30 views

How do you solve $\int_{-3}^{2} ( e^{-t+1} + sin (\frac{2\pi}{3}t) ) \delta(t- \frac{3}{2}) dt$?

How do you sovle the equation: $\int_{-3}^{2} ( e^{-t+1} + sin (\frac{2\pi}{3}t) ) \delta(t- \frac{3}{2}) dt$ Because of the $\delta(t- \frac{3}{2})$ this is only non-zero at $t=\frac{3}{2}$ But I ...
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2answers
37 views

How do I find the poles of this difference equation?

I have an equation: $$y(n) = 0.634x(n) - 0.634x(n-2) + 0.268y(n-2)$$ I completed a $z$ transform and got: $$ H(z) = \frac{1-0.268z^{-2}}{0.634 - 0.634z^{-2}}$$ What is the next step to find the ...
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37 views

Why does inputting complex exponentials into a system give its frequency response?

Let's say I have an FIR filter with the equation: $$ y[n] = \sum_{i=0}^{N-1} h[i] x[n-i] $$ I know that to find the frequency response of this filter, I need to input a complex exponential in place ...
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7 views

Systems properties assistance 2

I am a little confused about the properties (is it Linear, Causal, Time-Invariant, Stable?) of this T system. Would you tip me on this? enter image description here
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24 views

Calculating SNR

I am having trouble calculating SNR. We are given a wave file that we have to filter and then calculate the SNR before and after each filter and compare. I calculated the SNR using: ...
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1answer
46 views

Does the Discrete Fourier Transform assume a periodic signal, or one that dies off?

I keep hearing that the DFT assumes a periodic signal. E.g. the first answer in this MATLAB Q&A site. This doesn't make any sense to me. According to the derivations I've seen of the DFT one ...
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10 views

Does perlin noise have a constant-valued grid

As far as I understand, perlin noise is made by creating a grid, picking gradient vector over the vertices of the grid and computing the dot product of the distance vector from an end point to ...
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33 views

Can someone explain negative frequencies when doing the Fourier transform?

I apologize if this question has been asked before. I have looked and have not found a clear explanation. When doing the discrete Fourier transform (e.g. fft in MATLAB) for a vector of discrete time ...
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63 views

What is the inverse DTFT of a triangle in the frequency domain?

If I have a triangle in the frequency domain, what is the inverse DTFT? So $X(e^{jw})= 1-\frac{2*|w|}{\pi}$, what is x[n]?
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25 views

differentiating complex signal: dy/dx=d/dx(a+ib)= ???

1. Problem Outline I have a signal (array from 1:n) which has both real and imaginary parts. I require to take the derivative of the complex magnitude of that signal. The signal Y is a function of a ...
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12 views

Compute frequencies present in a sampled signal

I have a signal $$ X(t) = \sum^3_{k=1} A_k \cos (2\pi f_k t + \phi_k), -\infty < t < \infty $$ where $E[A_1^2] = 1, E[A_2^2] = 4, E[A_3^2] = 1$ and the phase functions are uniformly ...
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13 views

Minimizing function of overlapping volumes

I am implementing a method that performs alignment of slightly overlapping 3D volumes. To be more specific, I have a dataset of m x n volumes of size 1024 x 1024 x 100, and each volume overlaps for ...
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1answer
18 views

Systems properties assistance

I am a little confused about the properties (is it Linear, Causal, Time-Invariant, Stable?) of this T system. $$T[x(t)]=\sum_{k=t_0}^{t}{x(k)}$$ Some are obvious (it is linear), but can’t come up ...
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1answer
48 views

Why is the Discrete Fourier Transform exact for periodic signals?

I have in my course notes: $$ \sum_{k=0}^{N-1}y(k)e^{-j\omega_nk}\approx\sum_{k=-\infty}^{\infty}y(k)e^{-j\omega_nk} $$ Where the $\approx$ for some reason becomes $=$ when $y(k)$ is periodic. Could ...
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1answer
15 views

energy of a convolution

I have to find the energy of $y(t)$ $$h(t)=ho\;sinc^3(t/T)\\ x(t)=V_0+V_1\;sin(3\pi\; t/T)\\ y(t)=x*h\;(t) $$ Where "$*$" is the convolution and $sinc(t)=\frac {sin(\pi t)}{\pi t}$ I think that the ...
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1answer
24 views

$x(t)\rightarrow x(-t)$ and $x(t)\rightarrow x^\ast(-t)$ transforms

I have to determine if these transforms are linear and the core of the transforms: $x(t)\rightarrow x(-t), \quad t\in \mathbb{R}$ $x(t)\rightarrow x^\ast(-t), \quad t\in \mathbb{R}$ With "the ...
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0answers
21 views

Energy functions for CRF/MRF

I am currently working in image segmentation. I have read several papers and books where Markov or Conditional Random Fields are used in order to segment images. Most of them also mention an energy ...
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6 views

Independent components anaylsis with prior knowledge about the sources

I was wondering which algorithm performs the best if we already know something about the sources' distribution? Say all the sources(except the noise) are Bernoulli distributions. Which ...
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1answer
88 views

Expanding Fourier Series of $f(x)=\pi-x$ where $0<x<\pi$ (even and odd)

Please help me solve this Fourier series and correct my solution if it is wrong. it's a non-periodic function which we need to write its Fourier series (even and odd) : $ f(x)=\pi - x $ ; $ ...
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2answers
49 views

Time period of Sinusoidal Signal

I have confusion in calculating the Time period of different signals. 1) $x(t) =3cos(4t + \pi/3)$ solution: 2$\pi$/4 = $\pi$/2 (It is periodic) 3) $x(t) =cos(t-\pi/9) +9sin(2\pi t + \pi/12)$ Solution: ...
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20 views

How to calculate the cross-correlation of a halfwave?

I'm trying working on a vehicle modle and testing it with steering maneuvres standardised by the ISO. When it comes to data analysis the ISO standard says the following: 6.4 Time lag and Sine Time ...
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73 views

Sampling theorem - signal reconstruction

Hi and thank you for your help. I got stuck for the last days with this chapter in this book https://ccrma.stanford.edu/~jos/mdft/Sampling_Theorem.html. What I do not understand is ...
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26 views

Can someone explain the steps to the solution in detail please

I've got this question here. I already have the solution but I have no idea how the solution came up. Can someone share any steps in detail for dummies to understand the logic behind it. Any video any ...
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1answer
16 views

How to know for sure this is a low-pass filter

Given the transfer function $$H(z) = \frac{1}{4}(1+z^{-1}+z^{-2}+z^{-3})$$ I understand I have to switch $z$ to $e^{j\Omega}$, $0\le\Omega\le\pi$, in order to find out how it interacts with ...
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1answer
19 views

Constructing a 2 fold oversampled cosine basis in MATLAB

So I'm trying to construct a 2 fold oversampled cosine basis in MATLAB. I know how to construct the basis as a square matrix using the following command: $\mbox{dct(eye($n,n$))}$ where dct is the ...
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73 views

Best sources on complete transforms (classic orthonormal transforms) and overcomplete transforms in signal processing

In the introduction section of a thesis I read a little about classic orthonormal transforms such as Fourier, discrete cosine and wavelet transforms and their application in signal processing. Then ...
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34 views

Isomorphisms of normed vector spaces

I'm having difficulty with the following question: Show that, for $\mathbb{F} \in \{\mathbb{R}, \mathbb{C}\}$ and for an infinite discrete time-domain $\mathbb{T}$, $\exists$ an isomorphism of normed ...
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19 views

Is the sum of all cross-correlation samples representative of target existence likelihood?

Answers to this question take the peak in the cross correlation as the measure to the likelihood of the trigger signal exist in the received signal - this is pretty much text book. My question is ...
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19 views

2D convolution with one-dimensional function

I am somewhat stumped on what may be a very basic question. I have a 2D input function $F(x,y)$ and an impulse response $H(x)$ that is independent of $y$. The output function is a convolution ...